Lucas–Carmichael number

A Lucas- Carmichael number is a composite natural number which satisfies a similar condition as a Carmichael number.

Definition

A square-free odd natural number is called Lucas- Carmichael number if it has at least three prime factors, and applies to every prime divisor of the number: divides.

Example

3.7.19 = 399 and

Consequently, a 399 Lucas- Carmichael number.

The smallest Lucas- Carmichael numbers

The following numbers are Lucas- Carmichael numbers ( sequence A006972 in OEIS ):

The smallest Lucas- Carmichael number with five prime factors is 588 455 = 5.7.17.23.43.

Properties

Based on the identity n 1 = n / p 1 (p 1) * n / p holds for every prime factor p is a natural number n:

Thus an odd square-free number is accurate then n is a Lucas- Carmichael number if and only if for each of its prime divisors p 1 divides n / p - 1

There Fermat pseudoprimes under the Lucas- Carmichael numbers, but they are not a subset of Fermat pseudo- primes. It is not known whether a Lucas Carmichael number exists which is a Carmichael number simultaneously.

• Integer amount